Three Generations in $\mathfrak{e}_7$

It has long been a puzzling mystery in physics why quarks and leptons come in three distinct generations. These sets of particles appear identical in every way, except for their specific interactions with the Higgs boson.

Ideally, we would have a robust physical explanation for this. Failing that, a beautiful mathematical framework that makes this triplication feel natural would be a welcome substitute. This exploration represents my third foray into the intersection of the Standard Model and exceptional algebraic structures.

The Allure of Exceptional Structures

In the realm of algebra, when we categorize "famous gadgets"—such as simple Lie algebras, Euclidean Jordan algebras, or positive hermitian Jordan pairs—we typically find infinite series. However, these series are almost always accompanied by a few exceptional cases constructed using the octonions.

This recurring pattern is somewhat eerie. Some physicists suspect these exceptions aren't just mathematical curiosities but are fundamental to the universe. Perhaps the "quirks" of the Standard Model aren't random accidents, but natural consequences of an exceptional algebraic structure. While this is a long shot, the lack of progress in fundamental particle physics since the early 1980s makes such speculative paths worth pursuing.

The Evolution of the Theory

The journey toward linking these algebras to physics has progressed in stages:

From Jordan Algebras to Jordan Pairs

In 2018, Michel Dubois-Violette and Ivan Todorov discovered that the Standard Model gauge group emerges as the symmetries of the "exceptional Jordan algebra" and its internal ordinary Jordan algebras. I worked with John Baez and Paul Schwahn to clarify this, noting that these algebras arise naturally when axiomatizing quantum physics.

However, this only explained the gauge bosons (the photon, gluons, $W$, and $Z$ bosons), leaving the fermions (quarks and leptons) untouched.

Latham Boyle attempted to bridge this gap in 2020 by tensoring the exceptional Jordan algebra with complex numbers. Initially, this felt like a mere formal trick a disconnected mathematical maneuver. However, recent work by Endre Bokor, Latham Boyle, and myself has shown that the link to quantum physics remains intact if we use Jordan pairs (studied since 1975) instead of Jordan algebras.

The Algebraic Ladder

We discovered an "exceptional" Jordan pair built from bioctonions (octonions $\otimes$ complex numbers). This structure is linked to specific Lie algebras:

By moving from $\mathfrak{f}_4 \rightarrow \mathfrak{e}_6 \rightarrow \mathfrak{e}_7$, we increase the mathematical "room" available. $\mathfrak{e}_7$ is particularly tempting because the resulting Jordan pair possesses a distinct three-fold symmetry.

The $\mathfrak{e}_7$ Breakthrough

Around October 2025, I encountered a paper and thesis by Benjamin Nasmith: An exceptional combinatorial sequence and Standard Model particles (2020) and Tight Projective 5-Designs and Exceptional Structures. Nasmith claimed that three generations of fermions could fit within the Lie algebra $\mathfrak{e}_7$.

After translating these ideas into a more familiar mathematical language and expanding them, I produced the paper: Three generations in E7.

The Core Concept

The fundamental idea is as follows:

1. There is a standard way to embed the Standard Model gauge group Lie algebra, $\mathfrak{g}_{\text{SM}}$, into $\mathfrak{e}_7$.
2. We can construct an intermediate Lie algebra $\mathfrak{L}$ such that: $\mathfrak{L} = \mathfrak{g}_{\text{SM}} \oplus V$ where $V$ is a vector space of dimension $126$.
3. The action of $\mathfrak{g}_{\text{SM}}$ on $V$ (via the Lie bracket) mirrors exactly how it acts on three generations of Standard Model fermions and their antiparticles.

Key inclusions in $V$:

• Quarks and Leptons (3 generations)
• Antiparticles
• Right-handed neutrinos and their antiparticles

Current Limitations

It is important to emphasize that this is the observation of a fascinating mathematical pattern, not yet a complete physical theory.

What this model does NOT yet include:

• The spin of fermions and gauge bosons.
• The Higgs boson $\phi$ (though the paper discusses how it comes close).
• A Lagrangian $\mathcal{L}$.
• Predictions for particle masses or interactions.

% The relationship can be summarized as:
\mathfrak{g}_{SM} \subset \mathfrak{L} \subset \mathfrak{e}_7

While there is more to discuss—specifically the origin of this Lie algebra and the mysterious role of the right-handed neutrino—the pattern suggests that the triplication of generations may be an inherent property of $\mathfrak{e}_7$.